# The Noise-Sensitivity Phase Transition in Spectral Group Synchronization Over Compact Groups

## TL;DR

This paper rigorously analyzes the noise sensitivity of spectral group synchronization over compact groups, revealing a phase transition where the algorithm's success sharply changes with noise level.

## Contribution

It provides a theoretical phase transition threshold and exact formulas for accuracy, enhancing understanding of spectral synchronization's robustness under noise.

## Key findings

- Identifies a phase transition in noise sensitivity for spectral group synchronization.
- Derives asymptotically exact formulas for accuracy below the phase transition.
- Provides a risk estimate for practitioners to assess method performance.

## Abstract

In Group Synchronization, one attempts to find a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalization of a block matrix whose blocks are matrix representations of the measured pairwise differences. Assuming uniformly distributed measurement errors, we present a rigorous analysis of the accuracy and noise sensitivity of spectral group synchronization algorithms over any compact group, up to the rounding error. We identify a Baik-Ben Arous-P\'ech\'e type phase transition in the noise level, beyond which spectral group synchronization necessarily fails. Below the phase transition, spectral group synchronization succeeds in recovering the unknown group elements, but its performance deteriorates with the noise level. We provide asymptotically exact formulas for the accuracy of spectral group synchronization below the phase transition, up to the rounding error. We also provide a consistent risk estimate, allowing practitioners to estimate the method's accuracy from available measurements.